In a study of the word problem for groups, R.~J.~Thompson
considered a certain group $F$ of self-homeomorphisms of the Cantor
set and showed, among other things, that $F$ is finitely presented.
Using results of K.~S.~Brown and R.~Geoghegan, M.~N.~Dyer showed
that $F$ is the fundamental group of a finite two-complex $Z^2$
having Euler characteristic one and which is {\em Cockcroft}, in
the sense that each map of the two-sphere into $Z^2$ is
homologically trivial. We show that no proper covering complex of
$Z^2$ is Cockcroft. A general result on Cockcroft properties
implies that no proper regular covering complex of any finite
two-complex with fundamental group $F$ is Cockcroft.